Question

What is the domain and range of `1/(x-7)`?

Answer 1

Domain: all real numbers x such that `x != 7`
Range: all real numbers.

Explanation:

The domain is the set of all values of x such that the function is defined.

For this function, that's every value of x, with the exception of exactly 7, since that would lead to a division by zero.

The range is the set of all values y that can be produced by the function.

In this case, it's the set of all real numbers.

Mental experiment time:

Let x be just a TINY bit greater than 7. The denominator of your function is 7 minus that number, or just the tiny number.

1 divided by a tiny number is a BIG number. So you can make y = f(x) be a big as you want by choosing an input number x that is close to 7, but just a tiny bit greater than 7.

Now, make x be just a tiny bit LESS than 7. Now you have y equal to 1 divided by a very tiny NEGATIVE number. The result is a very large negative number. In fact you can make y = f(x) be as big a NEGATIVE number as you want by choosing an input number x that is close to 7, but just a tiny bit less.

Here's another sanity check: Graph the function... graph{1/(x-7) [-20, 20, -10, 10]}